Bit late to comment I know, but what are you saying won't match? You seem to be saying the OP is comparing a white LED spectrum captured with the PL Spectrometer to a capture with a commercial spectrometer of its tungsten calibration lamp.
I think the OP has a good point, as far as I know no-one has compared the spectrum from a well constructed PL Spectrometer and a commercial spectrometer both capturing the same source. Which might show that the PL Spec gives at least roughly right results or if they are in some way wildly different from a commercial spectrometer. I have noticed that the published spectra for 'white' LEDs show a narrow blue peak followed by a gentle yellow hump. Which is what you might expect for a white LED that has a blue LED plus a yellow phosphor. However if I capture the spectrum of a 'white' LED with the PL spectrometer I gat the sharp blue peak but the yellow hump has a pronounced dip in the middle. Either the published spectra for white LEDs are highly smoother / idealised or perhaps the transmission characteristics of the green and red filters in my spectrometers camera don't overlap and are attenuating the frequencies in the middle of the yellow hump.

Most spectra mismatches occur because of the light source. So, say you are trying to get the spectra of holmium oxide glass ( a common standard). That goes between the light source and the spectrometer. With common white led sources, since they dont provide light over the entire visible spectrum, there will be problems.

On the other hand, a tungsten lamp will provide relatively uniform light over the entire visible spectrum. So there will no problem acquiring data.

This doesnt eliminate problems with different scales. Different scales are used by different instruments ( absorbance, trsnsmittsnce, and a few less often used). These can drastically affect how a spectrum looks.